The First Steps in Algebra, by G. A. (George Albert) Wentworth doc

The sign − is read minus.Thus, 4− 3, read 4 minus 3, indicates that the number 3 is to besubtracted from the number 4, a− b, read a minus b, indicates thatthe number b is to be subtracte

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Title: The First Steps in Algebra

Author: G A (George Albert) Wentworth

Release Date: July 9, 2011 [EBook #36670]

Language: English

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*** START OF THIS PROJECT GUTENBERG EBOOK THE FIRST STEPS IN ALGEBRA ***

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FIRST STEPS IN ALGEBRA.

in the Office of the Librarian of Congress, at Washington.

All Rights Reserved.

Typography by J S Cushing & Co., Boston, U.S.A.

Presswork by Ginn & Co., Boston, U.S.A.

This book is written for pupils in the upper grades of mar schools and the lower grades of high schools The introduc-tion of the simple elements of Algebra into these grades will, it

gram-is thought, so stimulate the mental activity of the pupils, thatthey will make considerable progress in Algebra without detri-ment to their progress in Arithmetic, even if no more time isallowed for the two studies than is usually given to Arithmeticalone

The great danger in preparing an Algebra for very youngpupils is that the author, in endeavoring to smooth the path ofthe learner, will sacrifice much of the educational value of thestudy To avoid this real and serious danger, and at the sametime to gain the required simplicity, great care has been given

to the explanations of the fundamental operations and rules, thearrangement of topics, the model solutions of examples, and themaking of easy examples for the pupils to solve

Nearly all the examples throughout the book are new, andmade expressly for beginners

The first chapter clears the way for quite a full treatment

of simple integral equations with one unknown number In thefirst two chapters only positive numbers are involved, and thelearner is led to see the practical advantages of Algebra in itsmost interesting applications before he faces the difficulties ofnegative numbers

The third chapter contains a simple explanation of negativenumbers The recognition of the facts that the real nature of

iii

subtraction is counting backwards, and that the real nature ofmultiplication is forming the product from the multiplicand pre-cisely as the multiplier is formed from unity, makes an easy road

to the laws of addition and subtraction of algebraic numbers, and

to the law of signs in multiplication and division All the ciples and rules of this chapter are illustrated and enforced bynumerous examples involving simple algebraic expressions only.The ordinary processes with compound expressions, includ-ing simple cases of resolution into factors, and the treatment

prin-of fractions, naturally follow the third chapter The immediatesuccession of topics that require similar work is of the highestimportance to the beginner, and it is hoped that the half-dozenchapters on algebraic expressions will prove interesting, and givesufficient readiness in the use of symbols

A chapter on fractional equations with one unknown ber, a chapter on simultaneous equations with two unknownnumbers, and a chapter on quadratics follow in order Only onemethod of elimination is given in simultaneous equations andone method of completing the square in quadratics Moreover,the solution of the examples in quadratics requires the squareroots of only small numbers such as every pupil knows who haslearned the multiplication table In each of these three chapters

num-a considernum-able number of problems is given to stnum-ate num-and solve

By this means the learner is led to exercise his reasoning faculty,and to realize that the methods of Algebra require a strictly log-ical process These problems, however, are divided into classes,and a model solution of an example of each class is given as aguide to the solution of other examples of that class

The course may end with the chapter on quadratics, but the

simple questions of arithmetical progression and of geometricalprogression are so interesting in themselves, and show so clearlythe power of Algebra, that it will be a great loss not to take theshort chapters on these series.

The last chapter is on square and cube roots It is expectedthat pupils who use this book will learn how to extract thesquare and cube roots by the simple formulas of Algebra, and

be spared the necessity of committing to memory the long andtedious rules given in Arithmetic, rules that are generally for-gotten in less time than they are learned

Any corrections or suggestions will be thankfully received bythe author

A teachers’ edition is in press, containing solutions of amples, and such suggestions as experience with beginners hasshown to be valuable

ex-G A WENTWORTH.Exeter, NH, April, 1894

Chapter Page

I Introduction .1

II Simple Equations .24

III Positive and Negative Numbers .42

IV Addition and Subtraction 58

V Multiplication and Division .67

VI Special Rules in Multiplication and Division .82

VII Factors .92

VIII Common Factors and Multiples .110

IX Fractions .117

X Fractional Equations 136

XI Simultaneous Equations of the First Degree .162

XII Quadratic Equations 175

XIII Arithmetical Progression 189

XIV Geometrical Progression .197

XV Square and Cube Roots .203

Answers .221

vi

CHAPTER I.

INTRODUCTION

Note The principal definitions are put at the beginning of thebook for convenient reference They are not to be committed tomemory It is a good plan to have definitions and explanations readaloud in the class, and to encourage pupils to make comments uponthem, and ask questions about them

1 Algebra Algebra, like Arithmetic, treats of numbers

2 Units In counting separate objects or in measuringmagnitudes, the standards by which we count or measure arecalled units

Thus, in counting the boys in a school, the unit is a boy; inselling eggs by the dozen, the unit is a dozen eggs; in selling bricks

by the thousand, the unit is a thousand bricks; in measuring shortdistances, the unit is an inch, a foot, or a yard; in measuring longdistances, the unit is a rod or a mile

3 Numbers Repetitions of the unit are expressed by bers

num-4 Quantities A number of specified units of any kind iscalled a quantity; as, 4 pounds, 5 oranges

1

5 Number-Symbols in Arithmetic Arithmetic ploys the arbitrary symbols, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, calledfigures, to represent numbers.

em-6 Number-Symbols in Algebra Algebra employs theletters of the alphabet in addition to the figures of Arithmetic

to represent numbers Letters are used as general symbols ofnumbers to which any particular values may be assigned

PRINCIPAL SIGNS OF OPERATIONS

7 The signs of the fundamental operations are the same inAlgebra as in Arithmetic

8 The Sign of Addition, + The sign + is read plus.Thus, 4 + 3, read 4 plus 3, indicates that the number 3 is to

be added to the number 4, a + b, read a plus b, indicates that thenumber b is to be added to the number a

9 The Sign of Subtraction,− The sign − is read minus.Thus, 4− 3, read 4 minus 3, indicates that the number 3 is to besubtracted from the number 4, a− b, read a minus b, indicates thatthe number b is to be subtracted from the number a

10 The Sign of Multiplication, × The sign × is readtimes

Thus, 4× 3, read 4 times 3, indicates that the number 3 is to bemultiplied by 4, a× b, read a times b, indicates that the number b is

to be multiplied by the number a

A dot is sometimes used for the sign of multiplication Thus

2 · 3 · 4 · 5 means the same as 2 × 3 × 4 × 5 Either sign is readmultiplied by when followed by the multiplier $a × b, or $a · b,

is read a dollars multiplied by b

11 The Sign of Division, ÷ The sign ÷ is read divided

by

Thus, 4÷ 2, read 4 divided by 2, indicates that the number 4

is to be divided by 2, a÷ b, read a divided by b, indicates that thenumber a is to be divided by the number b

Division is also indicated by writing the dividend above thedivisor with a horizontal line between them

Thus, 4

2 means the same as 4÷ 2; a

b means the same as a÷ b

OTHER SIGNS USED IN ALGEBRA

12 The Sign of Equality, = The sign = is read is equal

to, when placed between two numbers and indicates that thesetwo numbers are equal

Thus, 8+4 = 12 means that 8+4 and 12 stand for equal numbers;

x + y = 20 means that x + y and 20 stand for equal numbers

13 The Sign of Inequality, > or < The sign > or <

is read is greater than and is less than respectively, and whenplaced between two numbers indicates that these two numbersare unequal, and that the number toward which the sign opens

is the greater

Thus, 9+6 > 12 means that 9+6 is greater than 12; and 9+6 < 16means that 9 + 6 is less than 16.

14 The Sign of Deduction, ∴ The sign ∴ is read hence

aggrega-Thus, each of the expressions a

+b

, a + b, (a + b), [a + b],{a + b},signifies that a + b is to be treated as a single number

FACTORS COEFFICIENTS POWERS

17 Factors When a number consists of the product oftwo or more numbers, each of these numbers is called a factor

of the product

The sign × is generally omitted between a figure and a letter,

or between letters; thus, instead of 63 × a × b, we write 63ab;instead of a × b × c, we write abc

The expression abc must not be confounded with a + b + c.abc is a product; a + b + c is a sum

but a+ b + c = 2 + 3 + 4 = 9

Note When a sign of operation is omitted in the notation ofArithmetic, it is always the sign of addition; but when a sign ofoperation is omitted in the notation of Algebra, it is always the sign

of multiplication Thus, 456 means 400 + 50 + 6, but 4ab means

4× a × b

18 Factors expressed by letters are called literal factors;factors expressed by figures are called numerical factors

19 If one factor of a product is equal to 0, the product

is equal to 0, whatever the values of the other factors Such afactor is called a zero factor

20 Coefficients A known factor of a product which isprefixed to another factor, to show the number of times thatfactor is taken, is called a coefficient

Thus, in 7c, 7 is the coefficient of c; in 7ax, 7 is the coefficient

of ax, or, if a is known, 7a is the coefficient of x

By coefficient, we generally mean the numerical coefficientwith its sign If no numerical coefficient is written, 1 is under-stood Thus, ax means the same as 1ax

21 Powers and Roots A product consisting of two ormore equal factors is called a power of that factor, and one

of the equal factors is called a root of the number

Thus, 9 = 3× 3; that is, 9 is a power of 3, and 3 is a root of 9

22 Indices or Exponents An index or exponent is anumber-symbol written at the right of, and a little above, anumber

If the index is a whole number, it shows the number of timesthe given number is taken as a factor.

Thus, a1, or simply a, denotes that a is taken once as a factor;

a2denotes that a is taken twice as a factor; a3denotes that a is takenthree times as a factor; and a4 denotes that a is taken four times as

a factor; and so on These are read: the first power of a; the secondpower of a; the third power of a; the fourth power of a; and so on

a3 is written instead of aaa

a4 is written instead of aaaa

23 The meaning of coefficient and exponent must be fully distinguished Thus,

ALGEBRAIC EXPRESSIONS.

24 An Algebraic Expression An algebraic expression is

a number written with algebraic symbols An algebraic sion may consist of one symbol, or of several symbols connected

expres-by signs

Thus, a, 3abc, 5a + 2b− 3c, are algebraic expressions

25 Terms A term is an algebraic expression, the parts ofwhich are not separated by the sign + or −

Thus, a, 5xy, 2ab×4cd, 3ab

4cdare algebraic expressions of one termeach A term may be separated into parts by the sign× or ÷

26 Simple Expressions An algebraic expression of oneterm is called a simple expression or monomial

Thus, 5xy, 7a× 2b, 7a ÷ 2b, are simple expressions

27 Compound Expressions An algebraic expression

of two or more terms is called a compound expression orpolynomial

Thus, 5xy + 7a, 2x− y − 3z, 4a − 3b + 2c − 3d are compoundexpressions

28 A polynomial of two terms is called a binomial; of threeterms, a trinomial

Thus, 3a− b is a binomial; and 3a − b + c is a trinomial

29 Positive and Negative Terms The terms of a pound expression preceded by the sign + are called positive

com-terms, and the terms preceded by the sign − are called tive terms The sign + before the first term is omitted.

nega-30 A positive and a negative term of the same numericalvalue cancel each other when combined

31 Like Terms Terms which have the same tion of letters are called like or similar terms; terms which donot have the same combination of letters are called unlike ordissimilar terms

combina-Thus, 5a2bc,−7a2bc, a2bc, are like terms; but 5a2bc, 5ab2c, 5abc2,are unlike terms

32 Degree of a Term A term that is the product of threeletters is said to be of the third degree; a term of four letters is

of the fourth degree; and so on

Thus, 5abc is of the third degree; 2a2b2c2, that is, 2aabbcc, is ofthe sixth degree

33 Degree of a Compound Expression The degree of acompound expression is the degree of that term of the expressionwhich is of the highest degree

Thus, a2x2+ bx + c is of the fourth degree, since a2x2 is of thefourth degree

34 Dominant Letter It often happens that there is oneletter in an expression of more importance than the rest, andthis is, therefore, called the dominant letter In such cases thedegree of the expression is generally called by the degree of thedominant letter

Thus, a2x2+ bx + c is of the second degree in x.

35 Arrangement of a Compound Expression A pound expression is said to be arranged according to the powers

com-of some letter when the exponents com-of that letter, reckoning fromleft to right, either descend or ascend in the order of magnitude.Thus, 3ax3−4bx2−6ax+8b is arranged according to the descend-ing powers of x, and 8b− 6ax − 4bx2+ 3ax3 is arranged according tothe ascending powers of x

The first process is represented by 10 + 3 + 2.

The second process is represented by 10 + (3 + 2)

If a man has 10 dollars and afterwards collects 3 dollars andthen pays a bill of 2 dollars, it makes no difference whether headds the 3 dollars collected to his 10 dollars and pays out of thissum his bill of 2 dollars, or pays the 2 dollars from the 3 dollarscollected and adds the remainder to his 10 dollars

The first process is represented by 10 + 3 − 2

The second process is represented by 10 + (3 − 2)

From (1) and (2) it follows that

If an expression within a parenthesis is preceded by the sign+,the parenthesis can be removed without making any change in thesigns of the expression

Conversely Any part of an expression can lie enclosedwithin a parenthesis and the sign + prefixed, without makingany change in the signs of the terms thus enclosed

38 Parentheses preceded by − If a man has 10 dollarsand has to pay two bills, one of 3 dollars and one of 2 dollars,

it makes no difference whether he takes 3 dollars and 2 dollars

in succession, or takes the 3 and 2 dollars at one time, from his

10 dollars

The first process is represented by 10 − 3 − 2

The second process is represented by 10 − (3 + 2)

If a man has 10 dollars consisting of 2 five-dollar bills, andhas a debt of 3 dollars to pay, he can pay his debt by giving afive-dollar bill and receiving 2 dollars

This process is represented by 10 − 5 + 2

Since the debt paid is 3 dollars, that is, (5 − 2) dollars, thenumber of dollars he has left can evidently be expressed by

10 − (5 − 2)

From (3) and (4) it follows that

If an expression within a parenthesis is preceded by the sign−,the parenthesis can be removed, provided the sign before eachterm within the parenthesis is changed, the sign+ to −, and thesign − to +

Conversely Any part of an expression can be enclosedwithin a parenthesis and the sign − prefixed, provided the sign

of each term enclosed is changed, the sign+ to −, and the sign −

to +

of dots underneath the first line and exactly similar to it.

• • • • • • • •

• • • • • • • •

• • • • • • • •

• • • • • • • •There are (5 + 3) dots in each line, and 4 lines The totalnumber of dots, therefore, is 4 × (5 + 3)

We see that in the left-hand group there are 4×5 dots, and inthe right-hand group 4 × 3 dots The sum of these two numbers(4 × 5) + (4 × 3) must be equal to the total number; that is,

4 × (8 − 3)

The total number of dots (crossed and not crossed) is (4×8),and the total number of dots crossed is (4 × 3) Therefore thetotal number of dots not crossed is

To multiply a compound expression by a simple one,

Multiply each term by the multiplier, and write the successiveproducts with the same signs as those of the original terms

1 If b = 4, find the value of 3b2.

If a = 5, b = 2, c = 0, x = 1, y = 3, find the value of

Find the value of each term, and combine the results

5ab stands for 5× 10 × 4 = 200;

op-When there is no sign expressed between single symbols orbetween simple and compound expressions, it must be remem-bered that the sign understood is the sign of multiplication.Thus 2(a − b) has the same meaning as 2 × (a − b)

2 Write six increased by four 6 + 4 Ans.

7 By how much does x exceed y?

8 Write four times three; the fourth power of three

4 × 3; 34 Ans

9 Write four times x; the fourth power of x

10 If one part of twenty-five is fifteen, what is the other part?

25 − 15 Ans

11 If one part of 35 is x, what is the other part?

12 If one part of x is a, what is the other part?

13 How much does ten lack of being twelve? 12 − 10 Ans

14 How much does x lack of being fourteen?

15 How much does x lack of being a?

16 If a man walks four miles an hour, how many miles will

17 If a man walks y miles an hour, how many miles will hewalk in x hours?

18 If a man walks y miles an hour, how many hours will ittake him to walk x miles?

Exercise 6.

1 If the dividend is twenty and the quotient five, what is

5 Ans

2 If the dividend is a and the quotient b, what is the divisor?

3 If John is twenty years old to-day, how old was he fouryears ago? How old will he be five years hence?

6 Write seven times the expression 2x minus y

7 Write the next integral number above four 4 + 1 Ans

8 If x is an integral number, write the next integral numberabove it; the next integral number below it

9 What number is less than 20 by d?

10 If the difference of two numbers is five, and the smallernumber is fifteen, what is the greater number? 15 + 5 Ans

11 If the difference of two numbers is eight, and the smallernumber is x, what is the greater number?

12 If the sum of two numbers is 30, and one of them is 20,

13 If the sum of two numbers is x, and one of them is 10,what is the other?

14 If 100 contains x ten times, what is the value of x?

3 What is the value of x if 7x equals 28?

4 If it takes 3 men 4 days to reap a field, how many dayswill it take one man to reap it? 3 × 4 Ans

5 If it takes a men b days to reap a field, how many dayswill it take one man to reap it?

6 What is the excess of 5x over 3x?

7 By how much does 20 − 3 exceed (10 + 1)?

20 − 3 − (10 + 1) Ans

8 By how much does 2x − 3 exceed (x + 1)?

9 If x stands for 10, find the value of 4(3x − 20)

10 If a stands for 10, and b for 2, find the value of 2(a − 2b).

11 How many cents in a dollars, b quarters, and c dimes?

12 A book-shelf contains French, Latin, and Greek books.There are 100 books in all, and there are x Latin and y Greekbooks How many French books are there?

13 A regiment of men is drawn up in 10 ranks of 80 meneach, and there are 15 men over How many men are there in

14 A regiment of men is drawn up in x ranks of y men each,and there are c men over How many men are there in theregiment?

Exercise 8

1 A room is 10 yards long and 8 yards wide In the middlethere is a carpet 6 yards square How many square yards ofoilcloth will be required to cover the rest of the floor?

10 × 8 − 62 Ans

2 A room is x yards long and y yards wide In the middlethere is a carpet a yards square How many square yards ofoilcloth will be required to cover the rest of the floor?

3 How many rolls of paper g feet long and k feet wide will berequired to paper a room, the perimeter of which, after properallowance is made for doors and windows, is p feet and the height

h feet?

4 Write six times the square of m, plus five c times theexpression d plus b minus a.

5 Write five times the expression two n plus one, diminished

by six times the expression c minus a plus b

6 A lady bought a dress for a dollars, a cloak for b dollars,two pairs of gloves for c dollars a pair She gave a hundred-dollarbill in payment How much money should be returned to her?

7 If a man can perform a piece of work in 4 days, how much

4 Ans

8 If a man can perform a piece of work in x days, how much

of it can he do in one day?

9 If A can do a piece of work in x days, B in y days, C in

z days, how much of it can they all do in one day, workingtogether?

10 Write an expression for the sum, and also for the product,

of three consecutive numbers of which the least is n

11 The product of two factors is 36; if one of the factors is x,what is the other factor?

12 If d is the divisor and q the quotient, what is the dividend?

13 If d is the divisor, q the quotient, and r the remainder,what is the dividend?

14 If x oranges can be bought for 50 cents, how many orangescan be bought for 100 cents?

15 What is the price in cents of x apples, if they are ten cents

a dozen?

16 If b oranges cost 6 cents, what will a oranges cost?

17 How many miles between two places, if a train travelling

m miles an hour requires 4 hours to make the journey?

18 If a man was x years old 10 years ago, how many yearsold will he be 7 years hence?

19 If a man was x years old y years ago, how many years oldwill he be c years hence?

20 If a floor is 3x yards long and 12 yards wide, how manysquare yards does the floor contain?

21 How many hours will it take to walk c miles, at the rate

of one mile in 15 minutes?

22 Write three consecutive numbers of which x is the middlenumber

23 If an odd number is represented by 2n + 1, what willrepresent the next odd number?

Thus, a + b = b + a, which is true for all values of a and b, is anidentical equation, and 3x + 2 = 8, which is true only when x standsfor 2, is an equation of condition.

For brevity, an identical equation is called an identity, and

an equation of condition is called simply an equation

46 We often employ an equation to discover an unknownnumber from its relation to known numbers We usually rep-resent the unknown number by one of the last letters of thealphabet, as x, y, z; and by way of distinction, we use the first

24

letters, a, b, c, etc., to represent numbers that are supposed

to be known, though not expressed in the number-symbols ofArithmetic

Thus, in the equation ax + b = c, x is supposed to represent anunknown number, and a, b, and c are supposed to represent knownnumbers

47 Simple Equations An equation which contains thefirst power of x, the symbol for the unknown number, and nohigher power, is called a simple equation, or an equation ofthe first degree

Thus, ax + b = c is a simple equation, or an equation of the firstdegree in x

48 Solution of an Equation To solve an equation is

to find the unknown number; that is, the number which, whensubstituted for its symbol in the given equation, renders theequation an identity This number is said to satisfy the equation,and is called the root of the equation

49 Axioms In solving an equation, we make use of thefollowing axioms:

Ax 1 If equal numbers be added to equal numbers, thesums will be equal

Ax 2 If equal numbers be subtracted from equal numbers,the remainders will be equal

Ax 3 If equal numbers be multiplied by equal numbers,the products will be equal

Ax 4 If equal numbers be divided by equal numbers, thequotients will be equal.

If, therefore, the two sides of an equation be increased by,diminished by, multiplied by, or divided by equal numbers, theresults will be equal

Thus, if 8x = 24, then 8x + 4 = 24 + 4, 8x − 4 = 24 − 4,

4× 8x = 4 × 24, and 8x ÷ 4 = 24 ÷ 4

50 Transposition of Terms It becomes necessary insolving an equation to bring all the terms that contain the sym-bol for the unknown number to one side of the equation, and allthe other terms to the other side This is called transposingthe terms We will illustrate by examples:

1 Find the number for which x stands when

9x + 11 − 11 = 70 + 11

2 Find the number for which x stands when x + b = a.

Subtract b from each side, x+ b − b = a − b (Ax 2)Since +b and −b in the left side cancel each other (§ 30), wehave x = a − b

3 Find the number for which x stands when x − b = a

The equation is x− b = a

Add +b to each side, x + b − b = a + b (Ax 1)Since +b and −b in the left side cancel each other (§ 30), wehave x = a + b

51 The effect of the operation in the preceding equations,when Axioms (1) and (2) are used, is to take a term from oneside and put it on the other side with its sign changed We canproceed in a like manner in any other case Hence the generalrule:

52 Any term may be transposed from one side of an tion to the other, provided its sign is changed

equa-53 Any term, therefore, which occurs on both sides withthe same sign may be removed from both without affecting theequality; and the sign of every term of an equation may bechanged without affecting the equality

54 Verification When the root is substituted for itssymbol in the given equation, and the equation reduces to anidentity, the root is said to be verified We will illustrate byexamples:

1 What number added to twice itself gives 24?

Let x stand for the number;

then 2x will stand for twice the number,

and the number added to twice itself will be x + 2x

But the number added to twice itself is 24

2 If 4x − 5 stands for 19, for what number does x stand?

We have the equation

4 Solve the equation

Remember that x must not be put for money, length, time,weight, etc., but for the required number of specified units ofmoney, length, time, weight, etc.

Express each statement carefully in algebraic language, andwrite out in full just what each expression stands for

Do not attempt to form the equation until all the statementsare made in symbols

We will illustrate by examples:

1 John has three times as many oranges as James, and theytogether have 32 How many has each?

Let x stand for the number of oranges James has;

then 3x is the number of oranges John has;

and x + 3x is the number of oranges they together have

We see that in the left-hand group there are 4×5 dots, and inthe right-hand group × dots The sum of these... expression within a parenthesis is preceded by the sign− ,the parenthesis can be removed, provided the sign before eachterm within the parenthesis is changed, the sign+ to −, and thesign − to +